Pengrti — Conformal Representation of Plane Curves. 661 
subtract these —— from the number previously ar¬ 
rived at. There are therefore at least 
p (p +1) (p 4- 2) 
6 
—5 (p— 1)— 
p {p ~ 2) (p — 3] 
or 
|-p 3 + 9p 2 —23p+15 
cubic relations among the F s independent of the quadratic re¬ 
lations. This result is in accordance with Weber for the case 
p = 4,* and it does not contradict Noether’s statement that 
there are + 2) — 5 (p — D since I say that there are 
at least — p 4 - 9 j> ?) —my formula always giving a 
smaller number than his. 
If p = 4 we select two functions belonging to the surface, 
one of weight twelve, the other of weight eighteen, the former 
of degree two, the latter of degree three in the 4>’s. For the 
normal curve representing F(x,y) = o we have then a twisted 
sextic in space of three dimensions and defined by the intersec¬ 
tion of these surfaces of second and third degree respectively 
in the Fs. Conversely, any twisted sextic which is the inter¬ 
section of such surfaces is the normal curve of some F (x, y) — o 
of deficiency four. For at some point of the common intersec¬ 
tion pass a plane tangent to the quadratic surface. It will cut 
the quadratic in two straight lines real or imaginary, and the 
cubic surface in a plane cubic. The lines each meet the cubic 
twice beside the original point. Projecting the twisted sextic 
from the original point on a plane we get a quintic with two 
double points which is a curve of deficiency 4 and hence the 
proposition is proved. 
If the quadric surface is an ellipsoid, by projecting the twisted 
sextic from the highest point the quintic obtained will have no 
infinite points. Its double points will be the projection of one 
real and one imaginary point of the sextic, and hence will look 
like an ordinary point on the curve. If the quadric be a 
cone one of the double points of the quintic may be real and if 
the surface be an hyperboloid or paraboloid both double points 
of the quintic may be real. If the quadric become a cone, by 
* Math. Ann., VoL XIII’, p. 47. 
